Chebyshev and The Fast Fourier Transform Methods for Signal Interpolation

TL;DR

This paper compares Chebyshev polynomial interpolation with Fourier methods for signal reconstruction, highlighting their mathematical foundations, advantages, and limitations in numerical approximation.

Contribution

It provides a detailed analysis of Chebyshev polynomial interpolation as an alternative to Fourier methods for signal approximation.

Findings

Chebyshev interpolation offers advantages in certain signal reconstruction scenarios.

Comparison shows differences in accuracy and computational efficiency between methods.

Mathematical formulations of Chebyshev polynomials are discussed in detail.

Abstract

Approximation theorem is one of the most important aspects of numerical analysis that has evolved over the years with many different approaches. Some of the most popular approximation methods include the Lebesgue approximation theorem, the Weierstrass approximation, and the Fourier approximation theorem. The limitations associated with various approximation methods are too crucial to ignore, and thus, the nature of a specific dataset may require using a specific approximation method for such estimates. In this report, we shall delve into Chebyshev's polynomials interpolation in detail as an alternative approach to reconstructing signals and compare the reconstruction to that of the Fourier polynomials. We will also explore the advantages and limitations of the Chebyshev polynomials and discuss in detail their mathematical formulation and equivalence to the cosine function over a given interval [a, b].